Just a few comments, as I don't have time at the moment to thoroughly read everything and perform simulations.
C13 should have been there to start off with. There's no reason why it shouldn't be there with a dual, rather than a single PSU and vice versa.
Why not use the LM358? Just saying it's crap for audio, without saying why isn't very helpful. The main three issues with the LM358 are: noise, crossover distortion and insufficient slew rate. These can be problems for an audio amplifier, but they're not always as bad as many people make out.
The LM358 has 40nV/√Hz of input voltage noise, so is no good for amplifying small, low level signals, such as those from a microphone. However isn't going to make much difference on a 400mV signal.
Crossover distortion has been covered before and can be cleared up by adding a pull-down resistor, as mentioned earlier. The LM358 may introduce other forms of distortion and I admit, I haven't done any tests or seen any experiments to quantify it. I suspect it will be more of a problem, at higher gains.
The LM358 only has a slew rate of 0.5V/μs or 500 000V/s. Try to get it to amplify a high frequency, high amplitude sinewave and it'll turn it into a triangle wave.
SR = 2xΠxfxVp
A 20kHz, 5Vp signal, will require an op-amp with a slew rate of at least 628 319V/s or 0.628319V/μs. In reality, you won't get an audio signal like that. Even a 10Vp signal which does have some content at 20kHz, won't have 5V of 20kHz in there, so it's a non-issue. Another thing is slew rate distortion doesn't sound nasty, like crossover distortion does, given the same figures of total harmonic distortion. Crossover distortion sounds very bad, while slew rate limiting distortion is very subtle, because it happens anyway when listening to sound, for a long distance, which attenuates the higher frequency content.
https://www.electronics-notes.com/articles/analogue_circuits/operational-amplifier-op-amp/slew-rate.phpTry putting stickers over the part numbers of op-amps, swapping them in the circuit. Look at the waveform on the oscilloscope and listen to it. Is there and audible difference?
Finally, as I said before, using constant current sources, rather than emitter resistors, in the differential pair will solve the problem varying power supply voltage, as well as boosting the gain and improving the common mode rejection ratio. I'll do a simulation if/when I get time.